A181433 Numbers k such that 11*k is 5 less than a square.

1, 4, 20, 29, 61, 76, 124, 145, 209, 236, 316, 349, 445, 484, 596, 641, 769, 820, 964, 1021, 1181, 1244, 1420, 1489, 1681, 1756, 1964, 2045, 2269, 2356, 2596, 2689, 2945, 3044, 3316, 3421, 3709, 3820, 4124, 4241, 4561, 4684, 5020, 5149, 5501, 5636, 6004

Offset

1,2

Comments

a(k)^k+1==0 (mod 3) for k of the form 3*(2*j+1); for other forms of k, a(k)^k-1==0 (mod 3). - Bruno Berselli, Oct 29 2010

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

G.f.: x*(1+3*x+14*x^2+3*x^3+x^4)/((1-x)^3*(1+x)^2). - Alexander R. Povolotsky, Oct 21 2010

a(n) = (22*n*(n-1) - 5*(2*n-1)*(-1)^n + 3)/8.

a(n) = a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5) for n>5, a(1)=1, a(2)=4, a(3)=20, a(4)=29, a(5)=61.

Sum_{i=1..n} a(i) = n*(22*n^2-15*(-1)^n-13)/24.

Mathematica

Sort[Flatten[ Table[{((7 + 11 k)^2 - 5)/11, ((4 + 11 k)^2 - 5)/11}, {k, 0, 20, 1}]]]
Select[Range[7000], IntegerQ[Sqrt[11#+5]]&] (* Harvey P. Dale, Nov 21 2014 *)

Prog

(Magma) &cat[ [((4+11*k)^2-5)/11, ((7+11*k)^2-5)/11] : k in [0..23] ]; // Klaus Brockhaus, Oct 20 2010
(PARI) x='x+O('x^50); Vec(x*(1+3*x+14*x^2+3*x^3+x^4)/((1-x)^3*(1+x)^2)) \\ G. C. Greubel, Feb 25 2017

Crossrefs

Sequence in context: A323040 A259755 A317249 * A079454 A205670 A115428

Adjacent sequences: A181430 A181431 A181432 * A181434 A181435 A181436

Keyword

nonn,easy

Author

Jon Perry, Oct 20 2010

Extensions

Formulas and more terms from Klaus Brockhaus and Bruno Berselli, Oct 20 2010

Sum added and superfluous formula removed by Bruno Berselli, Oct 22 2010 - Nov 15 2010

Status

approved